If you buy travel online, either on an airline website or one of the many online travel agencies such as Expedia and Priceline, you’ve probably seen what looks like a dubious pitch: “Book air and hotel together and save (fill in X amount of dollars).”

It takes some extra digging to see if the savings are real, but usually packages do indeed cost less than booking airfare and hotel separately. To prove that packages save money, you need to visit the airline’s website, find a fare for your dates of travel, and then move over to the hotel’s website to see what their lowest price is, then add the two amounts and compare the combined price to what the online travel agency would charge for a package on the same flights and the same room types.

Recently, I was booking a trip from New York to Detroit and did the math. On bookingbuddy.com I discovered that were I to buy a hotel plus air package my airfare on American Airlines would be thrown in for free. So I did some random searches on other itineraries and found that the deal was not a fluke.

Here are three recent examples I found:

———

Dallas to Los Angeles

Trip length: five nights, two guests

Airline: American

Hotel: Westin Los Angeles Airport

Airline web site price (AA.com): $362

Hotel web site price (starwood.com): $910

Air plus hotel bought separately: $,1272

Package price (Expedia.com) $1,198

Savings: $74

———

Los Angeles to Honolulu

Trip length: seven nights, two guests

Airline: Delta

Hotel: Hilton Waikiki Beach

Airline website price (Delta.com): $1,088

Hotel website price (hiltonwaikikibeach.com): $1,667 with Hilton Honors membership discount

Air plus hotel bought separately: $2,755

Package price (Travelocity.com): $2,325, with seventh night free promotion

Package savings: $430

———

Chicago to New York

Trip length: four nights, one guest

Airline: Delta

Hotel: Red Lion Inn and Suites Brooklyn

Airline web site price (Delta.com): $133

Hotel website price (redlion.com/Brooklyn): $845

Air plus hotel bought separately: $978

Package price (Priceline.com) $527

Savings: $451

A less-obvious additional advantage of packages: they sometimes come with more liberal rules and restrictions such as no required advance purchase. A colleague needed to make a last-minute trip from New York to Orlando, Fla., for a funeral and discovered that buying a package on jetBlue.com would cost far less than the last-minute airfare alone. “I don’t need the hotel since I’m staying with my mom,” he asked. “Will the hotel mind if I don’t show up?” He didn’t, and it didn’t, although I did advise him to cancel the hotel room on the day of arrival just as a courtesy.

Time and time again I read on travel blogs and “road warrior” websites, “Book directly with the hotel and airline!” There are some advantages to buying travel that way (you deal directly with the vendor instead of a third-party middleman, for one thing), but in many cases whatever you gain isn’t worth spending more than you need do. Package savings, along with the convenience of making a single purchase, are reasons why, despite efforts by hotels and airlines to entice direct bookings, online travel agencies still exist.